reserve V for Universe,
  a,b,x,y,z,x9,y9 for Element of V,
  X for Subclass of V,
  o,p,q,r,s,t,u,a1,a2,a3,A,B,C,D for set,
  K,L,M for Ordinal,
  n for Element of omega,
  fs for finite Subset of omega,
  e,g,h for Function,
  E for non empty set,
  f for Function of VAR,E,
  k,k1 for Element of NAT,
  v1,v2,v3 for Element of VAR,
  H,H9 for ZF-formula;

theorem Th10:
  X is closed_wrt_A1-A7 & a in Funcs(fs,omega) & b in X implies
  {a(#)x: x in b} in X
proof
  assume that
A1: X is closed_wrt_A1-A7 and
A2: a in Funcs(fs,omega) and
A3: b in X;
  Funcs(fs,omega) c= X by A1,Th8;
  then
A4: {a} in X by A1,A2,Th2;
  set A={a(#)x: x in b};
  set s={a};
  set B={y(#)x where y,x is Element of V: y in s & x in b};
A5: B c= A
  proof
    let q be object;
    assume q in B;
    then consider y,x such that
A6: q=y(#)x & y in s and
A7: x in b;
    q=a(#)x by A6,TARSKI:def 1;
    hence thesis by A7;
  end;
  A c= B
  proof
    let q be object;
    assume q in A;
    then
A8: ex x st q=a(#)x & x in b;
    a in s by TARSKI:def 1;
    hence thesis by A8;
  end;
  then
A9: A=B by A5;
  X is closed_wrt_A7 by A1;
  hence thesis by A3,A4,A9;
end;
